How do I solve the problem of positioning 11 pieces into a 8x8 puzzle?
Using your app, I was able to find a (spoiler alert!) solution manually. At least now you know your puzzle is solvable and you did not waste your money :)

It seems your app has a bug, though. I was unable to put the last piece, as shown in the picture. I was wondering if your solver, as it stands, will ever find a solution. 

Now the idea. It may be useful for your solver.

The board has 8x8=64 squares. Each piece will occupy 5 squares and you want to fit 11 pieces, so the final position will have 9 empty squares. Divide the board in two 8x4 rectangles, left and right. Now, it seems only fair that one rectangle should have 5 empty squares leaving the other with 4; with that in mind I've proceeded to fill the right part first and only then the left. After some trial and error, I got lucky.

I don't know if you can reach all solutions with this method. Notice that, in the solution given above, the bottom rectangle will end up with 6 empty squares. 

I don't know how to write an efficient solver either. For starters:


Build up a list of bad configurations: small sets of pieces/positions such that whenever you reach them, you know there is no hope.
At each step, put a piece in such way it minimizes the number of forced empty squares. 
A variant of the idea above: divide the board in four 4x4 parts; it seems reasonable each part should have at least one empty square but no so many; so, look for solutions forcing these parts to have (1,2,3,3) or (2,2,2,3) empty squares. 
Lookup whether this puzzle is known. Names that come to mind: Martin Gardner, Ian Stewart, Sam Lloyd. 


Can't say you will ever be able to see a list of all possible solutions. 

Nice puzzle, nice app that one you wrote, I've had a good time. Thank you.